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4 MAPPINGS 5

The product of three sets X, Y, Z is defined as X x Y x Z = (X x Y) x Z,
and the product of n sets is similarly defined by
induction: Xj x X₂ x • • • x X_n = (X_t x X₂ x • - • x X^) x X_n . An
element z of X₁ x • • - x X_n is written (jq, x₂ , . . . , x_n) instead of
((' **(*i> x₂\ ^3)5 • * • 9 *n-i)» *n)> •** *^s th^e ft*¹ Projection of z, and is written
#,• = pr_f z for 1 < z < ft. More generally, if i_l9 i₂ -> • • • » 4 ^are distinct indices
belonging to {1, 2, ...,«}, one writes

If X_t = X₂ = • • • = X_n = X we write X" instead of X x X x - • - x X n times.

4. MAPPINGS

Let X, Y be two sets, R(JC, y) a relation between x e X and y e Y; R is
said to be functional in y, if, for every x e X, there is one and only one y e Y
such that R(#, y) is true. The graph of such a relation is called & functional
graph in X x Y; such a subset F of X x Y is therefore characterized by
the fact that, for each x e X, there is one and only one y e Y such that
(x, y) eF; this element y is called the value of F at x, and written F(x).
A functional graph in X x Y in also called a mapping of X into Y, or & function
defined in X, taking its values in Y. It is customary, in the language, to talk
of a mapping and a functional graph as if they were two different kinds of
objects in one-to-one correspondence, and to speak therefore of " the graph
of a mapping," but this is a mere psychological distinction (corresponding
to whether one looks at F either "geometrically" or "analytically"). In
any case, it is fundamental, in modern mathematics, to get used to considering
a mapping as a single object, just as a point or a number, and to make a
clear distinction between the mapping F and any one of its values F(x);
the first is an element of ^3(X x Y), the second an element of Y, and one has
F = {(x, y) e X x Y \y = F(JC)}. The subsets of X x Y which have the property
of being functional graphs form a subset of ^5(X x Y), called the set of
mappings ofX into Y, and written Y^x or ^"(X, Y).

Examples of mappings

(1.4.1) If b is an element of Y, X x [b] is a functional graph, called the
constant mapping of X into Y, with the value b; it is essential to distinguish
it from the element b of Y.

(1.4.2) For Y = X, the relation y = x is functional in y; its graph is the set
of all pairs (x, x\ and is called the diagonal of X x X, or the identity mapping
of X into itself and is written 1 _v.



	4 MAPPINGS 5 The product of three sets X, Y, Z is defined as X x Y x Z = (X x Y) x Z, and the product of n sets is similarly defined by induction: Xj x X₂ x • • • x X_n = (X_t x X₂ x • - • x X^) x X_n . An element z of X₁ x • • - x X_n is written (jq, x₂ , . . . , x_n) instead of ((' *(i> x₂\ ^3)5 • * • 9 n-i)» n)> •** ^s th^e ft¹ Projection of z, and is written #,• = pr_f z for 1 < z < ft. More generally, if i_l9 i₂ -> • • • » 4 ^are distinct indices belonging to {1, 2, ...,«}, one writes

	If X_t = X₂ = • • • = X_n = X we write X" instead of X x X x - • - x X n times.

	4. MAPPINGS Let X, Y be two sets, R(JC, y) a relation between x e X and y e Y; R is said to be functional in y, if, for every x e X, there is one and only one y e Y such that R(#, y) is true. The graph of such a relation is called & functional graph in X x Y; such a subset F of X x Y is therefore characterized by the fact that, for each x e X, there is one and only one y e Y such that (x, y) eF; this element y is called the value of F at x, and written F(x). A functional graph in X x Y in also called a mapping of X into Y, or & function defined in X, taking its values in Y. It is customary, in the language, to talk of a mapping and a functional graph as if they were two different kinds of objects in one-to-one correspondence, and to speak therefore of " the graph of a mapping," but this is a mere psychological distinction (corresponding to whether one looks at F either "geometrically" or "analytically"). In any case, it is fundamental, in modern mathematics, to get used to considering a mapping as a single object, just as a point or a number, and to make a clear distinction between the mapping F and any one of its values F(x); the first is an element of ^3(X x Y), the second an element of Y, and one has F = {(x, y) e X x Y \y = F(JC)}. The subsets of X x Y which have the property of being functional graphs form a subset of ^5(X x Y), called the set of mappings ofX into Y, and written Y^x or ^"(X, Y). Examples of mappings (1.4.1) If b is an element of Y, X x [b] is a functional graph, called the constant mapping of X into Y, with the value b; it is essential to distinguish it from the element b of Y. (1.4.2) For Y = X, the relation y = x is functional in y; its graph is the set of all pairs (x, x\ and is called the diagonal of X x X, or the identity mapping of X into itself and is written 1 _v.